A tangent line L is drawn at the point (2, -4) on the parabola y2 = 8x . If the line L is also tangent to the circle x2 + y2 = a , then ‘a’ is equal to ____

The equation of tangent at (2, -4) to the parabola y2 = 8x is -4y = 4(x+2)⇒x+y+2=0If this line is a tangent to the circle x2 + y2 = a then a=21+1⇒a=2⇒a=2

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A tangent line L is drawn at the point (2, -4) on the parabola y² = 8x . If the line L is also tangent to the circle x² + y² = a , then ‘a’ is equal to ____

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answer is 2.

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Detailed Solution

The equation of tangent at (2, -4) to the parabola y² = 8x is -4y = 4(x+2) $\Rightarrow$ x+y+2=0
If this line is a tangent to the circle x² + y² = a then $\sqrt{a} = |\frac{2}{\sqrt{1 + 1}}| \Rightarrow \sqrt{a} = \sqrt{2} \Rightarrow a = 2$